Where does commensuration collide?

In mathematics, two magnitudes are commensurable when they share a common measure, a unit that divides both a whole number of times, so that their ratio is a ratio of whole numbers; when no such common unit exists, as with the side and diagonal of a square, they are incommensurable. Euclid's Elements, Book X, developed the classical theory of commensurable and incommensurable magnitudes, a discovery that forced Greek mathematics to confront quantities no common yardstick can capture. The referent is the existence or absence of a shared unit of measure between two magnitudes, decided by whether their ratio is rational. It is a property of magnitudes, distinct from paradigms that cannot be compared and from the social act of forcing things onto a common metric.[1]

Unit

Whether two magnitudes share a common measure, so that their ratio is that of whole numbers

Recovery means

Not applicable

Measured by

Decided by whether the ratio of the two magnitudes is rational, as when the Euclidean algorithm terminates

In the philosophy of science, incommensurability is Kuhn's thesis that rival scientific paradigms lack a common measure: because they define their key terms, problems, and standards differently, there is no neutral language or shared yardstick in which to fully translate one into the other and judge them point for point. Kuhn's 1962 The Structure of Scientific Revolutions used the term to explain why proponents of competing paradigms so often talk past each other during scientific revolutions. The referent is the absence of a shared standard for comparing whole systems of scientific thought, argued through the history of theory change. It is an epistemic relation between paradigms, borrowing the mathematicians' word for magnitudes with no common measure and applying it to bodies of knowledge.[2]

Unit

The absence of a common standard or language for fully comparing rival paradigms

Recovery means

Not applicable

Measured by

Argued from historical cases where competing paradigms redefine terms and standards and resist point-for-point translation

In sociology, commensuration is the social process of transforming different qualities into a common metric, so that dissimilar things, from a school's quality to a species' worth to a person's health, can be compared and traded off on a single scale such as a price, a ranking, or an index. Espeland and Stevens's 1998 Commensuration as a social process analysed this as a form of power that shapes what people attend to and value, absorbing information into a number while discarding context. The referent is the collective construction of a common measure across unlike things, studied through the metrics, rankings, and valuations that societies build. It is an achieved social practice, not a fixed property of magnitudes or an unbridgeable gap between paradigms.[3]

Unit

The transformation of distinct qualities onto a shared quantitative metric

Recovery means

Not applicable

Measured by

Studied through the rankings, prices, and indices that render unlike things comparable, and through what they omit